A particle moves under the influence of a central force, F r ( r ) = ? ? r ?3 .

Question

A particle moves under the influence of a central force, Fr(r) = ? ? r?3. At time t = 0, the radius = r0 and the velocity = v0 as shown in the figure.
(A) Determine the potential energy function U(r).
(B) Determine the velocity vc such that the particle will move on the circle.
(C) Now suppose v0 = vc /2. The particle will spiral into the origin. Calculate the time it will take the particle to reach r = 0.

A particle moves under the influence of a central force, Fr(r) = ? ? r?3. At time t = 0, the radius = r0 and the velocity = v0 as shown in the figure. Determine the potential energy function U(r). Determine the velocity vc such that the particle will move on the circle. Now suppose v0 = vc /2. The particle will spiral into the origin. Calculate the time it will take the particle to reach r = 0.

Explanation / Answer

http://www.chegg.com/homework-help/questions-and-answers/particle-moves-influence-central-force-fr-r-r-3-time-t-0-radius-r0-velocity-v0-shown-figur-q1019388

a) U = -int{F.dr} = -int{-alpha r^-3 dr} = -alpha/(2 r^2)

b) m (vc)^2/r = alpha r^(-3) ===> vc = sqrt(alpha/(m r^2))

c) v0 = vc/2 = 0.5 sqrt(alpha/(m r^2)) ===> t = d/v0 = r0/(0.5 sqrt(alpha/(m r^2)))

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